A KdV cellular automaton without integers

We show how to solve the ultradiscrete KdV equation when the initial condition takes real values. We do this by introducing a linear spectral problem for which we define dressing and undressing procedures that allow one to add a soliton to (or to subtract a soliton from) a solution of the ultradiscrete KdV equation. We demonstrate that an initial condition decomposes into a stable background (which may include the slowest solitons) and solitons that asymptotically detach themselves from this background. Our results show that the dynamics of the ultradiscrete KdV are quite rich and are on a par with those of its continuous and discrete counterparts.